On optimal ternary linear codes of dimension 6

Tatsuya Maruta; Yusuke Oya

doi:10.3934/amc.2011.5.505

Article Contents

2011, Volume 5, Issue 3: 505-520. Doi: 10.3934/amc.2011.5.505

This issue Previous Article On the order bounds for one-point AG codes Next Article New nearly optimal codebooks from relative difference sets

On optimal ternary linear codes of dimension 6

Tatsuya Maruta^1, and
Yusuke Oya^1,

1.
Department of Mathematics and Information Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531

Received: November 2010

Revised: April 2011

Published: August 2011

Abstract Related Papers Cited by

Abstract

We prove that $[g_3(6,d),6,d]_3$ codes for $d=253$-$267$ and $[g_3(6,d)+1,6,d]_3$ codes for $d=302, 303, 307$-$312$ exist and that $[g_3(6,d),6,d]_3$ codes for $d=175, 200, 302, 303, 308, 309$ and a $[g_3(6,133)+1,6,133]_3$ code do not exist, where $g_3(k,d)=\sum_{i=0}^{k-1} \lceil d / 3^i \rceil$. These determine $n_3(6,d)$ for $d=133, 175, 200, 253$-$267, 302, 303, 308$-$312$, where $n_q(k,d)$ is the minimum length $n$ for which an $[n,k,d]_q$ code exists. The updated $n_3(6,d)$ table is also given.

Keywords:

Mathematics Subject Classification: Primary: 94B27, 94B05; Secondary: 51E20, 05B25.

Citation:

Related Papers

Cited by

References

[1]	A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces, Geom. Dedicata, 9 (1980), 425-449.doi: 10.1007/BF00181559.
[2]	R. C. Bose and R. C. Burton, A characterization of flat spaces in a finite projective geometry and the uniqueness of the Hamming and the MacDonald codes, J. Combin. Theory, 1 (1966), 96-104.doi: 10.1016/S0021-9800(66)80007-8.
[3]	A. A. Bruen, Polynomial multiplicities over finite fields and intersection sets, J. Combin. Theory Ser. A, 60 (1992), 19-33.doi: 10.1016/0097-3165(92)90035-S.
[4]	M. van Eupen and R. Hill, An optimal ternary $[69,5,45]$ code and related codes, Des. Codes Cryptogr., 4 (1994), 271-282.doi: 10.1007/BF01388456.
[5]	M. van Eupen and P. Lisonêk, Classification of some optimal ternary linear codes of small length, Des. Codes Cryptogr., 10 (1997), 63-84.doi: 10.1023/A:1008292320488.
[6]	N. Hamada, A characterization of some $[n,k,d;q]$-codes meeting the Griesmer bound using a minihyper in a finite projective geometry, Discrete Math., 116 (1993), 229-268.doi: 10.1016/0012-365X(93)90404-H.
[7]	N. Hamada, The nonexistence of $[303,6,201;3]$-codes meeting the Griesmer bound, Technical Report OWUAM-009, Osaka Women's University, 1995.
[8]	N. Hamada and T. Helleseth, The uniqueness of $[87,5,57;3]$ codes and the nonexistence of $[258,6,171;3]$ codes, J. Statist. Plann. Inference, 56 (1996), 105-127.doi: 10.1016/S0378-3758(96)00013-4.
[9]	N. Hamada and T. Helleseth, The nonexistence of some ternary linear codes and update of the bounds for $n_3(6,d)$, $1\leq d\leq 243$, Math. Japon., 52 (2000), 31-43.
[10]	N. Hamada and T. Maruta, A survey of recent results on optimal linear codes and minihypers, unpublished manuscript, 2003.
[11]	U. Heim, On $t$-blocking sets in projective spaces, unpublished manuscript, 1994.
[12]	R. Hill, Caps and codes, Discrete Math., 22 (1978), 111-137.doi: 10.1016/0012-365X(78)90120-6.
[13]	R. Hill, Optimal linear codes, in "Cryptography and Coding II'' (ed. C. Mitchell), Oxford Univ. Press, Oxford, (1992), 75-104.
[14]	R. Hill, An extension theorem for linear codes, Des. Codes Cryptogr., 17 (1999), 151-157.doi: 10.1023/A:1008319024396.
[15]	R. Hill and D. E. Newton, Optimal ternary linear codes, Des. Codes Cryptogr., 2 (1992), 137-157.doi: 10.1007/BF00124893.
[16]	W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes," Cambridge University Press, Cambridge, 2003.
[17]	C. M. Jones, "Optimal Ternary Linear Codes," Ph.D thesis, University of Salford, 2000.
[18]	I. N. Landjev, The nonexistence of some optimal ternary linear codes of dimension five, Des. Codes Cryptogr., 15 (1998), 245-258.doi: 10.1023/A:1008317124941.
[19]	I. Landgev, T. Maruta and R. Hill, On the nonexistence of quaternary $[51,4,37]$ codes, Finite Fields Appl., 2 (1996), 96-110.
[20]	T. Maruta, On the nonexistence of $q$-ary linear codes ofdimension five, Des. Codes Cryptogr., 22 (2001), 165-177.
[21]	T. Maruta, Extendability of ternary linear codes, Des. Codes Cryptogr., 35 (2005), 175-190.doi: 10.1007/s10623-005-6400-7.
[22]	T. Maruta, "Griesmer Bound for Linear Codes over Finite Fields," available online at http://www.geocities.jp/mars39geo/griesmer.htm
[23]	Y. Oya, The nonexistence of $[132,6,86]_3$ codes and $[135,6,88]_3$ codes, Serdica J. Comput., to appear.
[24]	G. Pellegrino, Sul massimo ordine delle calotte in S_4,3, Matematiche (Catania), 25 (1970), 1-9.
[25]	M. Takenaka, K. Okamoto and T. Maruta, On optimal non-projective ternary linear codes, Discrete Math., 308 (2008), 842-854.doi: 10.1016/j.disc.2007.07.044.
[26]	H. N. Ward, Divisibility of codes meeting the Griesmer bound, J. Combin. Theory Ser. A, 83 (1998), 79-93.doi: 10.1006/jcta.1997.2864.
[27]	Y. Yoshida and T. Maruta, Ternary linear codes and quadrics, Electronic J. Combin., 16 (2009), 21 pp.